mirror of https://github.com/nmvdw/HITs-Examples
Add some properties from the Elephant
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@ -64,6 +64,29 @@ Section monad_fset.
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+ right. by apply HX1.
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Defined.
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Instance fmap_Surjection `{Univalence} {A B : Type} (f : A -> B)
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{Hsurj : IsSurjection f} : IsSurjection (fmap FSet f).
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Proof.
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apply BuildIsSurjection.
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hinduction; try (intros; apply path_ishprop).
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- apply tr. exists ∅. reflexivity.
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- intro b.
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specialize (Hsurj b).
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cbv in Hsurj.
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destruct Hsurj as [Hsurj _].
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strip_truncations.
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destruct Hsurj as [a Ha].
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apply tr.
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exists {|a|}. simpl. f_ap.
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- intros X Y HX HY.
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strip_truncations.
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apply tr.
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destruct HY as [Y' HY].
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destruct HX as [X' HX].
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exists (X' ∪ Y'). simpl.
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f_ap.
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Defined.
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Lemma bind_isIn `{Univalence} {A : Type} (X : FSet (FSet A)) (a : A) :
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(exists x, x ∈ X * a ∈ x) -> a ∈ (bind _ X).
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Proof.
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@ -85,6 +108,134 @@ Section monad_fset.
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exists z. split; assumption.
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Defined.
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(* So other properties of FSet(-) as acting on objects *)
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(* Elephant D Cor 5.4.5 *)
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Definition FSet_Unit_2 : FSet Unit -> Unit + Unit.
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Proof.
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hinduction.
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- left. apply tt.
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- intros []. right. apply tt.
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- intros A B.
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destruct A.
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+ destruct B.
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* left. apply tt.
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* right. apply tt.
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+ right. apply tt.
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- intros [[] | []] [[] | []] [[] | []]; reflexivity.
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- intros [[] | []] [[] | []]; reflexivity.
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- intros [[] | []]; reflexivity.
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- intros [[] | []]; reflexivity.
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- intros []; reflexivity.
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Defined.
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Definition Two_FSet_Unit : Unit + Unit -> FSet Unit.
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Proof.
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intros [[] | []].
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- exact ∅.
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- exact {|tt|}.
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Defined.
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Lemma FSet_Unit `{Funext} : FSet Unit <~> Unit + Unit.
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Proof.
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apply BuildEquiv with FSet_Unit_2.
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apply equiv_biinv_isequiv.
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split; exists Two_FSet_Unit.
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- intro x. hrecursion x.
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+ reflexivity.
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+ intros []. reflexivity.
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+ intros X Y HX HY.
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destruct (FSet_Unit_2 X) as [[] | []], (FSet_Unit_2 Y) as [[] | []];
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rewrite <- HX; rewrite <- HY; simpl.
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* apply (union_idem _)^.
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* apply (nl _)^.
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* apply (nr _)^.
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* apply (union_idem _)^.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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- intros [[] | []]; simpl; reflexivity.
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Defined.
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Definition fsum1 {A B : Type} : FSet (A + B) -> FSet A * FSet B.
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Proof.
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hrecursion.
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- exact (∅, ∅).
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- intros [a | b].
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+ exact ({|a|}, ∅).
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+ exact (∅, {|b|}).
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- intros [X Y] [X' Y'].
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exact (X ∪ X', Y ∪ Y').
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- intros [X X'] [Y Y'] [Z Z'].
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rewrite !assoc.
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reflexivity.
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- intros [X X'] [Y Y'].
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rewrite (comm Y X).
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rewrite (comm Y' X').
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reflexivity.
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- intros [X X'].
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rewrite !nl.
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reflexivity.
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- intros [X X'].
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rewrite !nr.
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reflexivity.
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- intros [a | b]; simpl; rewrite !union_idem; reflexivity.
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Defined.
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Definition fsum2 {A B : Type} : FSet A * FSet B -> FSet (A + B).
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Proof.
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intros [X Y].
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exact ((fset_fmap inl X) ∪ (fset_fmap inr Y)).
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Defined.
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Lemma fsum1_inl {A B : Type} (X : FSet A) :
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fsum1 (fset_fmap inl X) = (X, ∅ : FSet B).
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Proof.
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hinduction X; try reflexivity; try (intros; apply path_ishprop).
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intros X Y HX HY.
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rewrite HX, HY. simpl.
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rewrite nl.
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reflexivity.
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Defined.
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Lemma fsum1_inr {A B : Type} (Y : FSet B) :
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fsum1 (fset_fmap inr Y) = (∅ : FSet A, Y).
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Proof.
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hinduction Y; try reflexivity; try (intros; apply path_ishprop).
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intros X Y HX HY.
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rewrite HX, HY. simpl.
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rewrite nl.
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reflexivity.
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Defined.
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Lemma FSet_sum `{Funext} {A B : Type}: FSet (A + B) <~> FSet A * FSet B.
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Proof.
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apply BuildEquiv with fsum1.
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apply equiv_biinv_isequiv.
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split; exists fsum2.
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- intros X. hrecursion X; unfold fsum2; simpl.
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+ apply nl.
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+ intros [a | b]; simpl; [apply nr | apply nl].
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+ intros X Y HX HY.
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destruct (fsum1 X) as [X1 X2].
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destruct (fsum1 Y) as [Y1 Y2].
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rewrite (comm _ (fset_fmap inr Y2)).
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rewrite <-assoc.
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rewrite (assoc (fset_fmap inl Y1)).
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rewrite HY.
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rewrite (comm Y).
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rewrite assoc.
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rewrite HX.
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reflexivity.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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+ intros. apply path_ishprop.
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- intros [X Y]. simpl.
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rewrite !fsum1_inl, !fsum1_inr.
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simpl.
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rewrite nl, nr.
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reflexivity.
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Defined.
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End monad_fset.
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(** Lemmas relating operations to the membership predicate *)
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